Generalized Harmonic Numbers
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2018
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| _version_ | 1866910163358711808 |
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| author | Sousa, Jose Risomar |
| author_facet | Sousa, Jose Risomar |
| contents | This paper presents new formulae for the harmonic numbers of order $k$, $H_{k}(n)$, and for the partial sums of two Fourier series associated with them, denoted here by $C^m_{k}(n)$ and $S^m_{k}(n)$. I believe this new formula for $H_{k}(n)$ is an improvement over the digamma function, $ψ$, because it's simpler and it stems from Faulhaber's formula, which provides a closed-form for the sum of powers of the first $n$ positive integers. We demonstrate how to create an exact power series for the harmonic numbers, a new integral representation for $ζ(2k+1)$ and a new generating function for $ζ(2k+1)$, among many other original results. The approaches and formulae discussed here are entirely different from solutions available in the literature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1810_07877 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Generalized Harmonic Numbers Sousa, Jose Risomar Number Theory 11-XX This paper presents new formulae for the harmonic numbers of order $k$, $H_{k}(n)$, and for the partial sums of two Fourier series associated with them, denoted here by $C^m_{k}(n)$ and $S^m_{k}(n)$. I believe this new formula for $H_{k}(n)$ is an improvement over the digamma function, $ψ$, because it's simpler and it stems from Faulhaber's formula, which provides a closed-form for the sum of powers of the first $n$ positive integers. We demonstrate how to create an exact power series for the harmonic numbers, a new integral representation for $ζ(2k+1)$ and a new generating function for $ζ(2k+1)$, among many other original results. The approaches and formulae discussed here are entirely different from solutions available in the literature. |
| title | Generalized Harmonic Numbers |
| topic | Number Theory 11-XX |
| url | https://arxiv.org/abs/1810.07877 |